Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

and

for z > 0, respectively.

The above system of vector functions is also known as the system of localized vector spherical wave functions. Another system of vector functions which is suitable for analyzing axisymmetric particles with extreme geometries is the system of distributed vector spherical wave functions [49]. For an axisymmetric particle with the axis of rotation along the z-axis, the distributed vector spherical wave functions are defined as

where {zn}∞n=1 is a dense set of points on the z-axis (Fig. B.1), ez is the unit vector in the direction of the z-axis, n = 1, 2, ..., m Z, and l = 1 if m = 0 and l = 0, if m = 0. M1mn, Nmn1 is an entire solution to the Maxwell equations and M3mn, Nmn3 is a radiating solution to the Maxwell equations in R3 − {znez }. In the case of prolate scatterers, the distribution of the poles on the axis of rotation adequately describes the particle geometry. In contrast, it is clear from physical considerations that this arrangement is not suitable for oblate scatterers. In this case, the procedure of analytic continuation of the vector fields onto the complex plane along the source coordinate zn can be used

Fig. B.1. Sources distributed on the z-axis

270 B Wave Functions

Fig. B.2. Illustration of the complex plane. L is the generatrix of the surface and L is the image of L in the complex plane

(Fig. B.2). The complex plane Σ3 = {z3 = (Re z3, Im z3)/Re z3, Im z3 R} is the dual of the azimuthal plane ϕ = const., Σ = {η = (ρ, z)/ρ ≥ 0, z R}, and is defined by taking the real axis Re z3 along the z-axis. The vector spherical wave functions can be expressed in terms of the coordinates of the source point z3 Σ3 and the field point η Σ as

B.3 Rotations

We consider two coordinate systems Oxyz and Ox1y1z1 having the same origin. The coordinate system Ox1y1z1 is obtained by rotating the coordinate

Fig. B.3. Coordinate rotations

system Oxyz thought the Euler angles (α, β, γ) as shown in Fig. B.3. With (θ, ϕ) and (θ1, ϕ1) being the spherical angles of the same position vector r in the coordinate systems Oxyz and Ox1y1z1, the addition theorem for spherical wave functions under coordinate rotations is [58, 213, 239]

with the property ∆mm = ∆m m. The expression of Wigner d-functions for positive and negative values of the indices m and m is given by

272 B Wave Functions

and note that the above equation is valid for β < 0, if

The Wigner d-functions are real and have the following symmetry properties [58]:

Taking into account the above symmetry relations, we can express the d-

functions in terms of the d-functions with positive values of the indices m and m

The orthogonality property of the Wigner d-functions is similar to that of the associated Legendre functions and is given by

The Wigner d-functions are related to the generalized spherical functions Pmmn by the relation [78]

The generalized spherical functions are complex and have the following symmetry properties: